Hopf algebra equivariant cyclic homology and cyclic homology of crossed product algebras

Abstract

We introduce the cylindrical module A\H, where H is a Hopf algebra and A is a Hopf module algebra over H. We show that there exists an isomorphism between C ðA op z H cop Þ the cyclic module of the crossed product algebra A op z H cop , and DðA\HÞ, the cyclic module related to the diagonal of A\H. If S, the antipode of H, is invertible it follows that C ðA z HÞ F DðA op \H cop Þ. When S is invertible, we approximate HC ðA z HÞ by a spectral sequence and give an interpretation of E 0 ; E 1 and E 2 terms of this spectral sequence.

Equivariant Cyclic Cohomology of H-Algebras

“…An interesting feature in our generalization of Connes' Chern character which is patterned after Connes' original construction in [6] is the equivariant trace map Ψ (Proposition 5.2). We remark that the complex of cyclic equivariant cochains introduced in Section 3 is not quite the same as the complex that naturally appeared in [1]. We can prove, however, that it enjoys the same relation to crossed product algebras (Theorem 4.2).…”

“…Our cocyclic module in Theorem 3.1 is not quite the dual of the cyclic module that appeared in the E 2 -term of the spectral sequence in [1], but is very similar to it. In particular, Theorem 4.2 in the next section shows that this version of equivariant cyclic cohomology enjoys the same relation with cyclic cohomology of crossed product algebras as in the main theorem of [1]. The reason we prefer the present complex is that it works better for pairing with K-theory.…”

“…Let H be a Hopf algebra. We use Sweedler's notation and write ∆h = h (0) ⊗ h (1) , where summation is understood. Similarly, we write…”

“…One of the main themes studied in these papers is the relation between equivariant cyclic cohomology and the cyclic cohomology of the corresponding crossed product algebra. In [1], we extended one of the main results of these investigations, namely the Feigin-Tsygan and (independently) Nistor spectral sequence [7,15] to actions of Hopf algebras. The E 2 -term of this spectral sequence can be considered as the complex of noncommutative equivariant de Rham cochains on the given algebra.…”

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

“…In the general case the pertinent formulas are less obvious. Several approaches have been proposed in [AK1,AK2]. In the previous version of this paper we introduced equivariant cyclic cohomology in the case when the Hopf algebra admits a modular element, by which we mean a group-like element implementing the square of the antipode.…”